unadvisedefficiency
Here is an idea.
Make a list of all the apps you want to update with google play.
Create a new user profile, or use the Private Space if it is unused.
Install Google Play and log in with the same credentials as your main profile.
Install these apps into that profile. While you're at it, try ticking the box next to install that let's you install it on other devices too (which in this case will be your other profile). If you don't see this, don't worry.
If the install works, I believe it will alter the apps throughout the phone to have been installed by Google Play Store.
I don't know if it will work, but it might be worth a try?
Ohh and if it works, once all the apps have been switched to Google Play Store as their install method, feel free to log out of Google, then delete the profile.